Every undergraduate learns a fact about the symmetric group that $(1\,2)$ and $(1\,2\,\cdots\,n)$ generate $S_n$.
Also, it is interesting to know that for a prime $p$, any 2-cycle and any $p$-cycle generates $S_p$, but an arbitrary $2$-cycle and arbirtary $n$-cycle may not generate $S_n$. There is a criteria for the later:
Theorem: For $1\leq i<j\leq n$, $\langle (1\,2\,\cdots\,n), (i\,j)\rangle=S_n$ if and only if $(n,j-i)=1$.
With these interesting facts, I would like to ask two questions:
Q.1 If $\sigma,\tau\in S_n$ are cycles then what is the necessary and / or sufficient condition on $\sigma,\tau,n$ so that $\langle \sigma,\tau\rangle=S_n$?
Q.2 What is the totality of subsets $ S\subseteq S_n$ such that $|S|=2$ and $\langle S\rangle=S_n$?
(Answers to the both questions will involve combinatorics, and this will be a good place to see how combinatorics is useful to study finite groups; in fact, a part of answer will also involve nice combinatorial arguments. The second question may be hard!)